Certain properties of the power graph associated with a finite group

The power graph P(G) of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph P*(G) of P(G) is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph P(G) and its subgraph P*(G). We next provide necessary and sufficient conditions for a power graph P*(G) to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph P*(G) contains some cut-edges.